 # F-Test

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## What is an F Test?

An “F Test” is a catch-all term for any test that uses the F-distribution. In most cases, when people talk about the F-Test, what they are actually talking about is The F-Test to Compare Two Variances. However, the f-statistic is used in a variety of tests including regression analysis, the Chow test and the Scheffe Test (a post-hoc ANOVA test).

## General Steps for an F Test

If you’re running an F Test, you should use Excel, SPSS, Minitab or some other kind of technology to run the test. Why? Calculating the F test by hand, including variances, is tedious and time-consuming. Therefore you’ll probably make some errors along the way.

If you’re running an F Test using technology (for example, an F Test two sample for variances in Excel), the only steps you really need to do are Step 1 and 4 (dealing with the null hypothesis). Technology will calculate Steps 2 and 3 for you.

1. State the null hypothesis and the alternate hypothesis.
2. Calculate the F value. The F Value is calculated using the formula F = (SSE1 – SSE2 / m) / SSE2 / n-k, where SSE = residual sum of squares, m = number of restrictions and k = number of independent variables.
3. Find the F Statistic (the critical value for this test). The F statistic formula is:
F Statistic = variance of the group means / mean of the within group variances.
You can find the F Statistic in the F-Table.
4. Support or Reject the Null Hypothesis.

## F Test to Compare Two Variances

A Statistical F Test uses an F Statistic to compare two variances, s1 and s2, by dividing them. The result is always a positive number (because variances are always positive). The equation for comparing two variances with the f-test is:
F = s21 / s22

If the variances are equal, the ratio of the variances will equal 1. For example, if you had two data sets with a sample 1 (variance of 10) and a sample 2 (variance of 10), the ratio would be 10/10 = 1.

You always test that the population variances are equal when running an F Test. In other words, you always assume that the variances are equal to 1. Therefore, your null hypothesis will always be that the variances are equal.

## Assumptions

Several assumptions are made for the test. Your population must be approximately normally distributed (i.e. fit the shape of a bell curve) in order to use the test. Plus, the samples must be independent events. In addition, you’ll want to bear in mind a few important points:

## F Test to compare two variances by hand: Steps

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Warning: F tests can get really tedious to calculate by hand, especially if you have to calculate the variances. You’re much better off using technology (like Excel — see below).

These are the general steps to follow. Scroll down for a specific example (watch the video underneath the steps).

Step 1: If you are given standard deviations, go to Step 2. If you are given variances to compare, go to Step 3.

Step 2: Square both standard deviations to get the variances. For example, if σ1 = 9.6 and σ2 = 10.9, then the variances (s1 and s2) would be 9.62 = 92.16 and 10.92 = 118.81.

Step 3: Take the largest variance, and divide it by the smallest variance to get the f-value. For example, if your two variances were s1 = 2.5 and s2 = 9.4, divide 9.4 / 2.5 = 3.76.
Why? Placing the largest variance on top will force the F-test into a right tailed test, which is much easier to calculate than a left-tailed test.

Step 4: Find your degrees of freedom. Degrees of freedom is your sample size minus 1. As you have two samples (variance 1 and variance 2), you’ll have two degrees of freedom: one for the numerator and one for the denominator.

Step 5: Look at the f-value you calculated in Step 3 in the f-table. Note that there are several tables, so you’ll need to locate the right table for your alpha level. Unsure how to read an f-table? Read What is an f-table?.

Step 6: Compare your calculated value (Step 3) with the table f-value in Step 5. If the f-table value is smaller than the calculated value, you can reject the null hypothesis.

That’s it!

## Two Tailed F-Test

The difference between running a one or two tailed F test is that the alpha level needs to be halved for two tailed F tests. For example, instead of working at α = 0.05, you use α = 0.025; Instead of working at α = 0.01, you use α = 0.005.

With a two tailed F test, you just want to know if the variances are not equal to each other. In notation:
Ha = σ21 ≠ σ2 2

Example problem: Conduct a two tailed F Test on the following samples:
Sample 1: Variance = 109.63, sample size = 41.
Sample 2: Variance = 65.99, sample size = 21.

Step 1: Write your hypothesis statements:
Ho: No difference in variances.
Ha: Difference in variances.

Step 2: Calculate your F critical value. Put the highest variance as the numerator and the lowest variance as the denominator:
F Statistic = variance 1/ variance 2 = 109.63 / 65.99 = 1.66

Step 3: Calculate the degrees of freedom:
The degrees of freedom in the table will be the sample size -1, so:
Sample 1 has 40 df (the numerator).
Sample 2 has 20 df (the denominator).

Step 4: Choose an alpha level. No alpha was stated in the question, so use 0.05 (the standard “go to” in statistics). This needs to be halved for the two-tailed test, so use 0.025.

Step 5: Find the critical F Value using the F Table. There are several tables, so make sure you look in the alpha = .025 table. Critical F (40,20) at alpha (0.025) = 2.287. Step 6: Compare your calculated value (Step 2) to your table value (Step 5). If your calculated value is higher than the table value, you can reject the null hypothesis:
F calculated value: 1.66
F value from table: 2.287.
1.66 < 2 .287.
So we cannot reject the null hypothesis.

## F-Test to Compare Two Variances in Excel

Watch the video or read the steps below:

### F-test two sample for variances Excel 2013: Steps

Step 1: Click the “Data” tab and then click “Data Analysis.”
Step 2: Click “F test two sample for variances” and then click “OK.”
Step 3: Click the Variable 1 Range box and then type the location for your first set of data. For example, if you typed your data into cells A1 to A10, type “A1:A10” into that box.
Step 4: Click the Variable 2 box and then type the location for your second set of data. For example, if you typed your data into cells B1 to B10, type “B1:B10” into that box.
Step 5: Click the “Labels” box if your data has column headers.
Step 6: Choose an alpha level. In most cases, an alpha level of 0.05 is usually fine.
Step 7: Select a location for your output. For example, click the “New Worksheet” radio button.
Step 8: Click “OK.”
Step 9: Read the results. If your f-value is higher than your F critical value, reject the null hypothesis as your two populations have unequal variances.

Warning: Excel has a small “quirk.” Make sure that variance 1 is higher than variance 2. If it isn’t switch your input data around (i.e. make input 1 “B” and input 2 “A”). Otherwise, Excel will calculate an incorrect f-value. This is because the variance is a ratio of variance 1/variance 2, and Excel can’t work out which set of data is set 1 and set 2 without you explicitly telling it.

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## References

Archdeacon, T. (1994). Correlation and Regression Analysis: A Historian’s Guide. Univ of Wisconsin Press.

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